scholarly journals The Perturbed Riemann Problem for Special Keyfitz-Kranzer System with Three Piecewise Constant States

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Yuhao Jiang ◽  
Chun Shen
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Global Solutions ◽  
Wave Interactions ◽  
Small Perturbations ◽  
Riemann Solutions ◽  
Piecewise Constant

The Riemann problem for a special Keyfitz-Kranzer system is investigated and then seven different Riemann solutions are constructed. When the initial data are chosen as three piecewise constant states under suitable assumptions, the global solutions to the perturbed Riemann problem are constructed explicitly by studying all occurring wave interactions in detail. Furthermore, the stabilities of solutions are obtained under the specific small perturbations of Riemann initial data.

scholarly journals Construction of the Global Solutions to the Perturbed Riemann Problem for the Leroux System

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Pengpeng Ji ◽  
Chun Shen
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Phase Plane ◽  
Wave Interaction ◽  
Global Solutions ◽  
Geometrical Structures ◽  
Small Perturbations ◽  
Riemann Solutions ◽  
Piecewise Constant ◽  
Rarefaction Curves

The global solutions of the perturbed Riemann problem for the Leroux system are constructed explicitly under the suitable assumptions when the initial data are taken to be three piecewise constant states. The wave interaction problems are widely investigated during the process of constructing global solutions with the help of the geometrical structures of the shock and rarefaction curves in the phase plane. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.

scholarly journals Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations

2021 ◽  
Author(s):  
Yu Zhang ◽  
Yanyan Zhang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Global Solutions ◽  
Two Dimensional ◽  
Relativistic Euler Equations ◽  
Riemann Solutions ◽  
Isentropic Relativistic Euler Equations ◽  
The Stability

The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.

scholarly journals Four-Quadrant Riemann Problem for a 2×2 System II

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 592
Author(s):  
Jinah Hwang ◽  
Suyeon Shin ◽  
Myoungin Shin ◽  
Woonjae Hwang
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
Wave Interactions ◽  
Delta Shock ◽  
Elementary Wave ◽  
Initial Discontinuity ◽  
The Rich ◽  
Four Quadrant

In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions.

scholarly journals Analysis of the Stability of the Riemann Problem for a Simplified Model in Magnetogasdynamics

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Yujin Liu ◽  
Wenhua Sun
Keyword(s):  
Shock Wave ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Ideal Gas ◽  
Simplified Model ◽  
Characteristic Method ◽  
Small Perturbations ◽  
Riemann Solutions ◽  
One Dimensional ◽  
The Stability

The generalized Riemann problem for a simplified model of one-dimensional ideal gas in magnetogasdynamics in a neighborhood of the origin(t>0)in the(x,t)plane is considered. According to the different cases of the corresponding Riemann solutions, we construct the perturbed solutions uniquely with the characteristic method. We find that, for some case, the contact discontinuity appears after perturbation while there is no contact discontinuity of the corresponding Riemann solution. For most cases, the Riemann solutions are stable and the perturbation can not affect the corresponding Riemann solutions. While, for some few cases, the forward (backward) rarefaction wave can be transformed into the forward (backward) shock wave which shows that the Riemann solutions are unstable under such local small perturbations of the Riemann initial data.

scholarly journals The Perturbed Riemann Problem for the Aw-Rascle Model with Modified Chaplygin Gas Pressure

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Yujin Liu ◽  
Wenhua Sun
Keyword(s):  
Riemann Problem ◽  
Large Time ◽  
Chaplygin Gas ◽  
Global Structure ◽  
Gas Pressure ◽  
Modified Chaplygin Gas ◽  
Riemann Solutions ◽  
Asymptotic Behaviors ◽  
Piecewise Constant ◽  
The Stability

This paper is concerned with the perturbed Riemann problem for the Aw-Rascle model with the modified Chaplygin gas pressure. We obtain constructively the solutions when the initial values are three piecewise constant states. The global structure and the large-time asymptotic behaviors of the solutions are discussed case by case. Further, we obtain the stability of the corresponding Riemann solutions as the initial perturbed parameter tends to zero.

scholarly journals Interactions of Delta Shock Waves for Zero-Pressure Gas Dynamics with Energy Conservation Law

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Wei Cai ◽  
Yanyan Zhang
Keyword(s):  
Theoretical Analysis ◽  
Shock Waves ◽  
Initial Data ◽  
Gas Dynamics ◽  
Riemann Solutions ◽  
Piecewise Constant ◽  
Delta Shock ◽  
Vacuum States ◽  
Delta Shock Waves ◽  
Energy Conservation Law

We study the interactions of delta shock waves and vacuum states for the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. The Riemann problems with initial data of three piecewise constant states are solved case by case, and four different configurations of Riemann solutions are constructed. Furthermore, the numerical simulations completely coinciding with theoretical analysis are shown.

scholarly journals On the delta shockwave interactions for the isentropic Chaplygin gas system consisting of three scalar equations

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5355-5373 ◽  
Author(s):  
Meina Sun ◽  
Jie Xin
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Propagation Speed ◽  
Chaplygin Gas ◽  
Euler System ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Delta Shock ◽  
Scalar Equations

The Riemann problem for the one-dimensional version of isentropic compressible Euler system for the Chaplygin gas consisting of three scalar equations is considered. It is shown that the Riemann solutions involve only two situations: the combination of three contact discontinuities or a delta shock wave. The generalized Rankine-Hugoniot conditions of delta shock wave are derived and the exact delta shock wave solution including the strength and propagation speed is obtained explicitly. The solutions to the perturbed Riemann problem are constructed globally when the initial data are taken to be the three piecewise constant initial data. The wave interaction problem is extensively investigated and some interesting phenomena are observed. It is shown that the limits of solutions to the perturbed Riemann problem converge to the corresponding ones to the Riemann problem when the perturbation parameter tends to zero.

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